Properties

Label 6004.2.a.f.1.17
Level $6004$
Weight $2$
Character 6004.1
Self dual yes
Analytic conductor $47.942$
Analytic rank $1$
Dimension $25$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6004,2,Mod(1,6004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6004, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6004.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6004.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(47.9421813736\)
Analytic rank: \(1\)
Dimension: \(25\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.17
Character \(\chi\) \(=\) 6004.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.32467 q^{3} -0.559509 q^{5} +4.61440 q^{7} -1.24526 q^{9} +O(q^{10})\) \(q+1.32467 q^{3} -0.559509 q^{5} +4.61440 q^{7} -1.24526 q^{9} -1.22296 q^{11} -4.15292 q^{13} -0.741164 q^{15} -3.51585 q^{17} -1.00000 q^{19} +6.11255 q^{21} -0.0473115 q^{23} -4.68695 q^{25} -5.62355 q^{27} +2.43689 q^{29} +9.63624 q^{31} -1.62001 q^{33} -2.58180 q^{35} +0.465578 q^{37} -5.50124 q^{39} -10.4863 q^{41} -11.4150 q^{43} +0.696732 q^{45} -5.63127 q^{47} +14.2927 q^{49} -4.65733 q^{51} -0.488772 q^{53} +0.684255 q^{55} -1.32467 q^{57} -1.83018 q^{59} -1.23991 q^{61} -5.74611 q^{63} +2.32360 q^{65} +11.1066 q^{67} -0.0626720 q^{69} -14.2873 q^{71} +5.01017 q^{73} -6.20865 q^{75} -5.64321 q^{77} +1.00000 q^{79} -3.71357 q^{81} -14.2215 q^{83} +1.96715 q^{85} +3.22807 q^{87} -7.30400 q^{89} -19.1633 q^{91} +12.7648 q^{93} +0.559509 q^{95} -11.2966 q^{97} +1.52289 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9} - 3 q^{11} + q^{13} - 5 q^{15} - 13 q^{17} - 25 q^{19} - 24 q^{21} - 31 q^{23} + 21 q^{25} + 7 q^{27} - 19 q^{29} - 7 q^{31} - 30 q^{33} - q^{35} - 29 q^{37} - 26 q^{39} - 40 q^{41} - 40 q^{45} - 8 q^{47} - 9 q^{49} + 12 q^{51} - 38 q^{53} - 29 q^{55} - 4 q^{57} + 18 q^{59} - 26 q^{61} - 40 q^{63} - 70 q^{65} - 13 q^{67} + q^{69} - 47 q^{71} - 8 q^{73} + 7 q^{75} - 19 q^{77} + 25 q^{79} - 19 q^{81} - 8 q^{83} - 33 q^{85} - 50 q^{87} - 54 q^{89} - 12 q^{91} - 24 q^{93} + 8 q^{95} - 4 q^{97} - 39 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.32467 0.764797 0.382399 0.923997i \(-0.375098\pi\)
0.382399 + 0.923997i \(0.375098\pi\)
\(4\) 0 0
\(5\) −0.559509 −0.250220 −0.125110 0.992143i \(-0.539928\pi\)
−0.125110 + 0.992143i \(0.539928\pi\)
\(6\) 0 0
\(7\) 4.61440 1.74408 0.872040 0.489434i \(-0.162797\pi\)
0.872040 + 0.489434i \(0.162797\pi\)
\(8\) 0 0
\(9\) −1.24526 −0.415085
\(10\) 0 0
\(11\) −1.22296 −0.368735 −0.184367 0.982857i \(-0.559024\pi\)
−0.184367 + 0.982857i \(0.559024\pi\)
\(12\) 0 0
\(13\) −4.15292 −1.15181 −0.575907 0.817515i \(-0.695351\pi\)
−0.575907 + 0.817515i \(0.695351\pi\)
\(14\) 0 0
\(15\) −0.741164 −0.191368
\(16\) 0 0
\(17\) −3.51585 −0.852718 −0.426359 0.904554i \(-0.640204\pi\)
−0.426359 + 0.904554i \(0.640204\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 6.11255 1.33387
\(22\) 0 0
\(23\) −0.0473115 −0.00986512 −0.00493256 0.999988i \(-0.501570\pi\)
−0.00493256 + 0.999988i \(0.501570\pi\)
\(24\) 0 0
\(25\) −4.68695 −0.937390
\(26\) 0 0
\(27\) −5.62355 −1.08225
\(28\) 0 0
\(29\) 2.43689 0.452519 0.226259 0.974067i \(-0.427350\pi\)
0.226259 + 0.974067i \(0.427350\pi\)
\(30\) 0 0
\(31\) 9.63624 1.73072 0.865360 0.501151i \(-0.167090\pi\)
0.865360 + 0.501151i \(0.167090\pi\)
\(32\) 0 0
\(33\) −1.62001 −0.282007
\(34\) 0 0
\(35\) −2.58180 −0.436404
\(36\) 0 0
\(37\) 0.465578 0.0765405 0.0382703 0.999267i \(-0.487815\pi\)
0.0382703 + 0.999267i \(0.487815\pi\)
\(38\) 0 0
\(39\) −5.50124 −0.880904
\(40\) 0 0
\(41\) −10.4863 −1.63768 −0.818840 0.574021i \(-0.805383\pi\)
−0.818840 + 0.574021i \(0.805383\pi\)
\(42\) 0 0
\(43\) −11.4150 −1.74077 −0.870385 0.492371i \(-0.836130\pi\)
−0.870385 + 0.492371i \(0.836130\pi\)
\(44\) 0 0
\(45\) 0.696732 0.103863
\(46\) 0 0
\(47\) −5.63127 −0.821405 −0.410702 0.911770i \(-0.634716\pi\)
−0.410702 + 0.911770i \(0.634716\pi\)
\(48\) 0 0
\(49\) 14.2927 2.04182
\(50\) 0 0
\(51\) −4.65733 −0.652157
\(52\) 0 0
\(53\) −0.488772 −0.0671380 −0.0335690 0.999436i \(-0.510687\pi\)
−0.0335690 + 0.999436i \(0.510687\pi\)
\(54\) 0 0
\(55\) 0.684255 0.0922650
\(56\) 0 0
\(57\) −1.32467 −0.175457
\(58\) 0 0
\(59\) −1.83018 −0.238269 −0.119134 0.992878i \(-0.538012\pi\)
−0.119134 + 0.992878i \(0.538012\pi\)
\(60\) 0 0
\(61\) −1.23991 −0.158754 −0.0793769 0.996845i \(-0.525293\pi\)
−0.0793769 + 0.996845i \(0.525293\pi\)
\(62\) 0 0
\(63\) −5.74611 −0.723942
\(64\) 0 0
\(65\) 2.32360 0.288207
\(66\) 0 0
\(67\) 11.1066 1.35688 0.678441 0.734655i \(-0.262656\pi\)
0.678441 + 0.734655i \(0.262656\pi\)
\(68\) 0 0
\(69\) −0.0626720 −0.00754482
\(70\) 0 0
\(71\) −14.2873 −1.69559 −0.847795 0.530324i \(-0.822070\pi\)
−0.847795 + 0.530324i \(0.822070\pi\)
\(72\) 0 0
\(73\) 5.01017 0.586396 0.293198 0.956052i \(-0.405280\pi\)
0.293198 + 0.956052i \(0.405280\pi\)
\(74\) 0 0
\(75\) −6.20865 −0.716913
\(76\) 0 0
\(77\) −5.64321 −0.643104
\(78\) 0 0
\(79\) 1.00000 0.112509
\(80\) 0 0
\(81\) −3.71357 −0.412619
\(82\) 0 0
\(83\) −14.2215 −1.56101 −0.780507 0.625147i \(-0.785039\pi\)
−0.780507 + 0.625147i \(0.785039\pi\)
\(84\) 0 0
\(85\) 1.96715 0.213367
\(86\) 0 0
\(87\) 3.22807 0.346085
\(88\) 0 0
\(89\) −7.30400 −0.774222 −0.387111 0.922033i \(-0.626527\pi\)
−0.387111 + 0.922033i \(0.626527\pi\)
\(90\) 0 0
\(91\) −19.1633 −2.00886
\(92\) 0 0
\(93\) 12.7648 1.32365
\(94\) 0 0
\(95\) 0.559509 0.0574045
\(96\) 0 0
\(97\) −11.2966 −1.14700 −0.573499 0.819206i \(-0.694415\pi\)
−0.573499 + 0.819206i \(0.694415\pi\)
\(98\) 0 0
\(99\) 1.52289 0.153056
\(100\) 0 0
\(101\) 14.6789 1.46060 0.730302 0.683124i \(-0.239379\pi\)
0.730302 + 0.683124i \(0.239379\pi\)
\(102\) 0 0
\(103\) 3.26007 0.321224 0.160612 0.987018i \(-0.448653\pi\)
0.160612 + 0.987018i \(0.448653\pi\)
\(104\) 0 0
\(105\) −3.42003 −0.333761
\(106\) 0 0
\(107\) −9.22541 −0.891854 −0.445927 0.895069i \(-0.647126\pi\)
−0.445927 + 0.895069i \(0.647126\pi\)
\(108\) 0 0
\(109\) 5.88996 0.564156 0.282078 0.959391i \(-0.408976\pi\)
0.282078 + 0.959391i \(0.408976\pi\)
\(110\) 0 0
\(111\) 0.616736 0.0585380
\(112\) 0 0
\(113\) 2.70267 0.254246 0.127123 0.991887i \(-0.459426\pi\)
0.127123 + 0.991887i \(0.459426\pi\)
\(114\) 0 0
\(115\) 0.0264712 0.00246845
\(116\) 0 0
\(117\) 5.17145 0.478101
\(118\) 0 0
\(119\) −16.2235 −1.48721
\(120\) 0 0
\(121\) −9.50438 −0.864035
\(122\) 0 0
\(123\) −13.8908 −1.25249
\(124\) 0 0
\(125\) 5.41994 0.484774
\(126\) 0 0
\(127\) 3.12290 0.277113 0.138556 0.990355i \(-0.455754\pi\)
0.138556 + 0.990355i \(0.455754\pi\)
\(128\) 0 0
\(129\) −15.1211 −1.33134
\(130\) 0 0
\(131\) 4.71439 0.411898 0.205949 0.978563i \(-0.433972\pi\)
0.205949 + 0.978563i \(0.433972\pi\)
\(132\) 0 0
\(133\) −4.61440 −0.400120
\(134\) 0 0
\(135\) 3.14643 0.270802
\(136\) 0 0
\(137\) −9.04312 −0.772606 −0.386303 0.922372i \(-0.626248\pi\)
−0.386303 + 0.922372i \(0.626248\pi\)
\(138\) 0 0
\(139\) −15.4865 −1.31355 −0.656774 0.754087i \(-0.728080\pi\)
−0.656774 + 0.754087i \(0.728080\pi\)
\(140\) 0 0
\(141\) −7.45956 −0.628208
\(142\) 0 0
\(143\) 5.07884 0.424714
\(144\) 0 0
\(145\) −1.36346 −0.113229
\(146\) 0 0
\(147\) 18.9331 1.56158
\(148\) 0 0
\(149\) −4.93131 −0.403988 −0.201994 0.979387i \(-0.564742\pi\)
−0.201994 + 0.979387i \(0.564742\pi\)
\(150\) 0 0
\(151\) −12.7734 −1.03948 −0.519742 0.854323i \(-0.673972\pi\)
−0.519742 + 0.854323i \(0.673972\pi\)
\(152\) 0 0
\(153\) 4.37813 0.353951
\(154\) 0 0
\(155\) −5.39157 −0.433061
\(156\) 0 0
\(157\) 14.1468 1.12904 0.564520 0.825419i \(-0.309061\pi\)
0.564520 + 0.825419i \(0.309061\pi\)
\(158\) 0 0
\(159\) −0.647461 −0.0513470
\(160\) 0 0
\(161\) −0.218314 −0.0172056
\(162\) 0 0
\(163\) 9.46483 0.741342 0.370671 0.928764i \(-0.379128\pi\)
0.370671 + 0.928764i \(0.379128\pi\)
\(164\) 0 0
\(165\) 0.906411 0.0705640
\(166\) 0 0
\(167\) −9.66094 −0.747586 −0.373793 0.927512i \(-0.621943\pi\)
−0.373793 + 0.927512i \(0.621943\pi\)
\(168\) 0 0
\(169\) 4.24676 0.326674
\(170\) 0 0
\(171\) 1.24526 0.0952271
\(172\) 0 0
\(173\) 20.8594 1.58591 0.792955 0.609281i \(-0.208542\pi\)
0.792955 + 0.609281i \(0.208542\pi\)
\(174\) 0 0
\(175\) −21.6275 −1.63488
\(176\) 0 0
\(177\) −2.42438 −0.182227
\(178\) 0 0
\(179\) 8.43749 0.630648 0.315324 0.948984i \(-0.397887\pi\)
0.315324 + 0.948984i \(0.397887\pi\)
\(180\) 0 0
\(181\) −14.8275 −1.10212 −0.551059 0.834466i \(-0.685776\pi\)
−0.551059 + 0.834466i \(0.685776\pi\)
\(182\) 0 0
\(183\) −1.64246 −0.121414
\(184\) 0 0
\(185\) −0.260495 −0.0191520
\(186\) 0 0
\(187\) 4.29973 0.314427
\(188\) 0 0
\(189\) −25.9493 −1.88754
\(190\) 0 0
\(191\) 8.10583 0.586517 0.293259 0.956033i \(-0.405260\pi\)
0.293259 + 0.956033i \(0.405260\pi\)
\(192\) 0 0
\(193\) −15.6151 −1.12400 −0.562000 0.827137i \(-0.689968\pi\)
−0.562000 + 0.827137i \(0.689968\pi\)
\(194\) 0 0
\(195\) 3.07800 0.220420
\(196\) 0 0
\(197\) 16.3818 1.16716 0.583579 0.812056i \(-0.301652\pi\)
0.583579 + 0.812056i \(0.301652\pi\)
\(198\) 0 0
\(199\) −2.15732 −0.152928 −0.0764640 0.997072i \(-0.524363\pi\)
−0.0764640 + 0.997072i \(0.524363\pi\)
\(200\) 0 0
\(201\) 14.7125 1.03774
\(202\) 0 0
\(203\) 11.2448 0.789229
\(204\) 0 0
\(205\) 5.86717 0.409781
\(206\) 0 0
\(207\) 0.0589149 0.00409487
\(208\) 0 0
\(209\) 1.22296 0.0845936
\(210\) 0 0
\(211\) −3.15067 −0.216901 −0.108451 0.994102i \(-0.534589\pi\)
−0.108451 + 0.994102i \(0.534589\pi\)
\(212\) 0 0
\(213\) −18.9259 −1.29678
\(214\) 0 0
\(215\) 6.38680 0.435576
\(216\) 0 0
\(217\) 44.4655 3.01852
\(218\) 0 0
\(219\) 6.63681 0.448474
\(220\) 0 0
\(221\) 14.6010 0.982172
\(222\) 0 0
\(223\) 15.7673 1.05585 0.527927 0.849290i \(-0.322969\pi\)
0.527927 + 0.849290i \(0.322969\pi\)
\(224\) 0 0
\(225\) 5.83645 0.389097
\(226\) 0 0
\(227\) −20.7491 −1.37716 −0.688582 0.725159i \(-0.741766\pi\)
−0.688582 + 0.725159i \(0.741766\pi\)
\(228\) 0 0
\(229\) 14.5450 0.961161 0.480581 0.876951i \(-0.340426\pi\)
0.480581 + 0.876951i \(0.340426\pi\)
\(230\) 0 0
\(231\) −7.47538 −0.491844
\(232\) 0 0
\(233\) 19.4721 1.27566 0.637829 0.770178i \(-0.279833\pi\)
0.637829 + 0.770178i \(0.279833\pi\)
\(234\) 0 0
\(235\) 3.15075 0.205532
\(236\) 0 0
\(237\) 1.32467 0.0860464
\(238\) 0 0
\(239\) 18.6860 1.20870 0.604348 0.796720i \(-0.293434\pi\)
0.604348 + 0.796720i \(0.293434\pi\)
\(240\) 0 0
\(241\) −11.7621 −0.757662 −0.378831 0.925466i \(-0.623674\pi\)
−0.378831 + 0.925466i \(0.623674\pi\)
\(242\) 0 0
\(243\) 11.9514 0.766683
\(244\) 0 0
\(245\) −7.99691 −0.510904
\(246\) 0 0
\(247\) 4.15292 0.264244
\(248\) 0 0
\(249\) −18.8388 −1.19386
\(250\) 0 0
\(251\) −0.835572 −0.0527408 −0.0263704 0.999652i \(-0.508395\pi\)
−0.0263704 + 0.999652i \(0.508395\pi\)
\(252\) 0 0
\(253\) 0.0578598 0.00363762
\(254\) 0 0
\(255\) 2.60582 0.163183
\(256\) 0 0
\(257\) −8.84356 −0.551646 −0.275823 0.961208i \(-0.588950\pi\)
−0.275823 + 0.961208i \(0.588950\pi\)
\(258\) 0 0
\(259\) 2.14836 0.133493
\(260\) 0 0
\(261\) −3.03455 −0.187834
\(262\) 0 0
\(263\) −11.3052 −0.697111 −0.348555 0.937288i \(-0.613328\pi\)
−0.348555 + 0.937288i \(0.613328\pi\)
\(264\) 0 0
\(265\) 0.273473 0.0167993
\(266\) 0 0
\(267\) −9.67537 −0.592123
\(268\) 0 0
\(269\) 16.2614 0.991473 0.495736 0.868473i \(-0.334898\pi\)
0.495736 + 0.868473i \(0.334898\pi\)
\(270\) 0 0
\(271\) 9.68191 0.588134 0.294067 0.955785i \(-0.404991\pi\)
0.294067 + 0.955785i \(0.404991\pi\)
\(272\) 0 0
\(273\) −25.3849 −1.53637
\(274\) 0 0
\(275\) 5.73193 0.345648
\(276\) 0 0
\(277\) 21.8545 1.31311 0.656553 0.754280i \(-0.272014\pi\)
0.656553 + 0.754280i \(0.272014\pi\)
\(278\) 0 0
\(279\) −11.9996 −0.718396
\(280\) 0 0
\(281\) −7.71302 −0.460120 −0.230060 0.973176i \(-0.573892\pi\)
−0.230060 + 0.973176i \(0.573892\pi\)
\(282\) 0 0
\(283\) −11.5401 −0.685985 −0.342993 0.939338i \(-0.611441\pi\)
−0.342993 + 0.939338i \(0.611441\pi\)
\(284\) 0 0
\(285\) 0.741164 0.0439028
\(286\) 0 0
\(287\) −48.3879 −2.85625
\(288\) 0 0
\(289\) −4.63882 −0.272872
\(290\) 0 0
\(291\) −14.9643 −0.877222
\(292\) 0 0
\(293\) −32.4078 −1.89329 −0.946643 0.322285i \(-0.895549\pi\)
−0.946643 + 0.322285i \(0.895549\pi\)
\(294\) 0 0
\(295\) 1.02400 0.0596197
\(296\) 0 0
\(297\) 6.87736 0.399065
\(298\) 0 0
\(299\) 0.196481 0.0113628
\(300\) 0 0
\(301\) −52.6734 −3.03604
\(302\) 0 0
\(303\) 19.4447 1.11707
\(304\) 0 0
\(305\) 0.693739 0.0397234
\(306\) 0 0
\(307\) 16.0863 0.918094 0.459047 0.888412i \(-0.348191\pi\)
0.459047 + 0.888412i \(0.348191\pi\)
\(308\) 0 0
\(309\) 4.31851 0.245671
\(310\) 0 0
\(311\) −1.86961 −0.106016 −0.0530080 0.998594i \(-0.516881\pi\)
−0.0530080 + 0.998594i \(0.516881\pi\)
\(312\) 0 0
\(313\) 21.9742 1.24205 0.621027 0.783789i \(-0.286716\pi\)
0.621027 + 0.783789i \(0.286716\pi\)
\(314\) 0 0
\(315\) 3.21500 0.181145
\(316\) 0 0
\(317\) 27.3044 1.53357 0.766783 0.641906i \(-0.221856\pi\)
0.766783 + 0.641906i \(0.221856\pi\)
\(318\) 0 0
\(319\) −2.98021 −0.166860
\(320\) 0 0
\(321\) −12.2206 −0.682088
\(322\) 0 0
\(323\) 3.51585 0.195627
\(324\) 0 0
\(325\) 19.4645 1.07970
\(326\) 0 0
\(327\) 7.80224 0.431465
\(328\) 0 0
\(329\) −25.9849 −1.43260
\(330\) 0 0
\(331\) −20.6619 −1.13568 −0.567841 0.823138i \(-0.692221\pi\)
−0.567841 + 0.823138i \(0.692221\pi\)
\(332\) 0 0
\(333\) −0.579763 −0.0317708
\(334\) 0 0
\(335\) −6.21422 −0.339519
\(336\) 0 0
\(337\) −33.6223 −1.83152 −0.915761 0.401723i \(-0.868411\pi\)
−0.915761 + 0.401723i \(0.868411\pi\)
\(338\) 0 0
\(339\) 3.58014 0.194447
\(340\) 0 0
\(341\) −11.7847 −0.638177
\(342\) 0 0
\(343\) 33.6516 1.81701
\(344\) 0 0
\(345\) 0.0350656 0.00188787
\(346\) 0 0
\(347\) −20.2019 −1.08450 −0.542248 0.840219i \(-0.682427\pi\)
−0.542248 + 0.840219i \(0.682427\pi\)
\(348\) 0 0
\(349\) 22.7121 1.21575 0.607875 0.794033i \(-0.292022\pi\)
0.607875 + 0.794033i \(0.292022\pi\)
\(350\) 0 0
\(351\) 23.3542 1.24655
\(352\) 0 0
\(353\) 18.8430 1.00291 0.501455 0.865184i \(-0.332798\pi\)
0.501455 + 0.865184i \(0.332798\pi\)
\(354\) 0 0
\(355\) 7.99388 0.424271
\(356\) 0 0
\(357\) −21.4908 −1.13741
\(358\) 0 0
\(359\) −21.9617 −1.15910 −0.579548 0.814938i \(-0.696771\pi\)
−0.579548 + 0.814938i \(0.696771\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −12.5901 −0.660811
\(364\) 0 0
\(365\) −2.80324 −0.146728
\(366\) 0 0
\(367\) −9.89811 −0.516677 −0.258338 0.966054i \(-0.583175\pi\)
−0.258338 + 0.966054i \(0.583175\pi\)
\(368\) 0 0
\(369\) 13.0581 0.679777
\(370\) 0 0
\(371\) −2.25539 −0.117094
\(372\) 0 0
\(373\) −23.4530 −1.21435 −0.607175 0.794568i \(-0.707697\pi\)
−0.607175 + 0.794568i \(0.707697\pi\)
\(374\) 0 0
\(375\) 7.17962 0.370754
\(376\) 0 0
\(377\) −10.1202 −0.521217
\(378\) 0 0
\(379\) 14.3601 0.737629 0.368815 0.929503i \(-0.379764\pi\)
0.368815 + 0.929503i \(0.379764\pi\)
\(380\) 0 0
\(381\) 4.13681 0.211935
\(382\) 0 0
\(383\) 11.3562 0.580274 0.290137 0.956985i \(-0.406299\pi\)
0.290137 + 0.956985i \(0.406299\pi\)
\(384\) 0 0
\(385\) 3.15743 0.160918
\(386\) 0 0
\(387\) 14.2146 0.722568
\(388\) 0 0
\(389\) −11.2439 −0.570087 −0.285043 0.958515i \(-0.592008\pi\)
−0.285043 + 0.958515i \(0.592008\pi\)
\(390\) 0 0
\(391\) 0.166340 0.00841217
\(392\) 0 0
\(393\) 6.24499 0.315018
\(394\) 0 0
\(395\) −0.559509 −0.0281520
\(396\) 0 0
\(397\) 5.51636 0.276858 0.138429 0.990372i \(-0.455795\pi\)
0.138429 + 0.990372i \(0.455795\pi\)
\(398\) 0 0
\(399\) −6.11255 −0.306010
\(400\) 0 0
\(401\) −29.3633 −1.46633 −0.733166 0.680050i \(-0.761958\pi\)
−0.733166 + 0.680050i \(0.761958\pi\)
\(402\) 0 0
\(403\) −40.0185 −1.99347
\(404\) 0 0
\(405\) 2.07778 0.103246
\(406\) 0 0
\(407\) −0.569381 −0.0282232
\(408\) 0 0
\(409\) 19.6435 0.971307 0.485653 0.874152i \(-0.338582\pi\)
0.485653 + 0.874152i \(0.338582\pi\)
\(410\) 0 0
\(411\) −11.9791 −0.590887
\(412\) 0 0
\(413\) −8.44518 −0.415560
\(414\) 0 0
\(415\) 7.95707 0.390597
\(416\) 0 0
\(417\) −20.5145 −1.00460
\(418\) 0 0
\(419\) −18.9281 −0.924700 −0.462350 0.886697i \(-0.652994\pi\)
−0.462350 + 0.886697i \(0.652994\pi\)
\(420\) 0 0
\(421\) −0.538550 −0.0262473 −0.0131237 0.999914i \(-0.504178\pi\)
−0.0131237 + 0.999914i \(0.504178\pi\)
\(422\) 0 0
\(423\) 7.01236 0.340953
\(424\) 0 0
\(425\) 16.4786 0.799329
\(426\) 0 0
\(427\) −5.72143 −0.276879
\(428\) 0 0
\(429\) 6.72777 0.324820
\(430\) 0 0
\(431\) 11.8396 0.570292 0.285146 0.958484i \(-0.407958\pi\)
0.285146 + 0.958484i \(0.407958\pi\)
\(432\) 0 0
\(433\) 1.18547 0.0569700 0.0284850 0.999594i \(-0.490932\pi\)
0.0284850 + 0.999594i \(0.490932\pi\)
\(434\) 0 0
\(435\) −1.80613 −0.0865975
\(436\) 0 0
\(437\) 0.0473115 0.00226321
\(438\) 0 0
\(439\) −38.8818 −1.85573 −0.927863 0.372921i \(-0.878356\pi\)
−0.927863 + 0.372921i \(0.878356\pi\)
\(440\) 0 0
\(441\) −17.7981 −0.847528
\(442\) 0 0
\(443\) 39.6056 1.88172 0.940859 0.338798i \(-0.110020\pi\)
0.940859 + 0.338798i \(0.110020\pi\)
\(444\) 0 0
\(445\) 4.08665 0.193726
\(446\) 0 0
\(447\) −6.53234 −0.308969
\(448\) 0 0
\(449\) −18.9386 −0.893767 −0.446883 0.894592i \(-0.647466\pi\)
−0.446883 + 0.894592i \(0.647466\pi\)
\(450\) 0 0
\(451\) 12.8242 0.603870
\(452\) 0 0
\(453\) −16.9205 −0.794995
\(454\) 0 0
\(455\) 10.7220 0.502656
\(456\) 0 0
\(457\) −3.53642 −0.165427 −0.0827133 0.996573i \(-0.526359\pi\)
−0.0827133 + 0.996573i \(0.526359\pi\)
\(458\) 0 0
\(459\) 19.7716 0.922857
\(460\) 0 0
\(461\) −14.2480 −0.663598 −0.331799 0.943350i \(-0.607656\pi\)
−0.331799 + 0.943350i \(0.607656\pi\)
\(462\) 0 0
\(463\) 18.3849 0.854421 0.427210 0.904152i \(-0.359496\pi\)
0.427210 + 0.904152i \(0.359496\pi\)
\(464\) 0 0
\(465\) −7.14204 −0.331204
\(466\) 0 0
\(467\) −16.7556 −0.775355 −0.387678 0.921795i \(-0.626723\pi\)
−0.387678 + 0.921795i \(0.626723\pi\)
\(468\) 0 0
\(469\) 51.2501 2.36651
\(470\) 0 0
\(471\) 18.7399 0.863487
\(472\) 0 0
\(473\) 13.9600 0.641883
\(474\) 0 0
\(475\) 4.68695 0.215052
\(476\) 0 0
\(477\) 0.608646 0.0278680
\(478\) 0 0
\(479\) −27.9174 −1.27558 −0.637788 0.770212i \(-0.720151\pi\)
−0.637788 + 0.770212i \(0.720151\pi\)
\(480\) 0 0
\(481\) −1.93351 −0.0881604
\(482\) 0 0
\(483\) −0.289194 −0.0131588
\(484\) 0 0
\(485\) 6.32057 0.287002
\(486\) 0 0
\(487\) 7.68782 0.348368 0.174184 0.984713i \(-0.444271\pi\)
0.174184 + 0.984713i \(0.444271\pi\)
\(488\) 0 0
\(489\) 12.5378 0.566977
\(490\) 0 0
\(491\) −14.4279 −0.651123 −0.325562 0.945521i \(-0.605553\pi\)
−0.325562 + 0.945521i \(0.605553\pi\)
\(492\) 0 0
\(493\) −8.56773 −0.385871
\(494\) 0 0
\(495\) −0.852072 −0.0382978
\(496\) 0 0
\(497\) −65.9274 −2.95725
\(498\) 0 0
\(499\) −24.9439 −1.11664 −0.558320 0.829626i \(-0.688554\pi\)
−0.558320 + 0.829626i \(0.688554\pi\)
\(500\) 0 0
\(501\) −12.7975 −0.571752
\(502\) 0 0
\(503\) −35.7299 −1.59312 −0.796558 0.604562i \(-0.793348\pi\)
−0.796558 + 0.604562i \(0.793348\pi\)
\(504\) 0 0
\(505\) −8.21298 −0.365473
\(506\) 0 0
\(507\) 5.62554 0.249839
\(508\) 0 0
\(509\) 18.1728 0.805493 0.402747 0.915311i \(-0.368056\pi\)
0.402747 + 0.915311i \(0.368056\pi\)
\(510\) 0 0
\(511\) 23.1189 1.02272
\(512\) 0 0
\(513\) 5.62355 0.248286
\(514\) 0 0
\(515\) −1.82404 −0.0803768
\(516\) 0 0
\(517\) 6.88679 0.302881
\(518\) 0 0
\(519\) 27.6318 1.21290
\(520\) 0 0
\(521\) −8.14603 −0.356884 −0.178442 0.983950i \(-0.557106\pi\)
−0.178442 + 0.983950i \(0.557106\pi\)
\(522\) 0 0
\(523\) 14.6470 0.640471 0.320235 0.947338i \(-0.396238\pi\)
0.320235 + 0.947338i \(0.396238\pi\)
\(524\) 0 0
\(525\) −28.6492 −1.25035
\(526\) 0 0
\(527\) −33.8795 −1.47582
\(528\) 0 0
\(529\) −22.9978 −0.999903
\(530\) 0 0
\(531\) 2.27904 0.0989018
\(532\) 0 0
\(533\) 43.5487 1.88630
\(534\) 0 0
\(535\) 5.16171 0.223160
\(536\) 0 0
\(537\) 11.1769 0.482318
\(538\) 0 0
\(539\) −17.4794 −0.752890
\(540\) 0 0
\(541\) −20.3124 −0.873298 −0.436649 0.899632i \(-0.643835\pi\)
−0.436649 + 0.899632i \(0.643835\pi\)
\(542\) 0 0
\(543\) −19.6415 −0.842897
\(544\) 0 0
\(545\) −3.29549 −0.141163
\(546\) 0 0
\(547\) −7.70587 −0.329479 −0.164740 0.986337i \(-0.552678\pi\)
−0.164740 + 0.986337i \(0.552678\pi\)
\(548\) 0 0
\(549\) 1.54400 0.0658963
\(550\) 0 0
\(551\) −2.43689 −0.103815
\(552\) 0 0
\(553\) 4.61440 0.196224
\(554\) 0 0
\(555\) −0.345069 −0.0146474
\(556\) 0 0
\(557\) 11.4245 0.484071 0.242035 0.970267i \(-0.422185\pi\)
0.242035 + 0.970267i \(0.422185\pi\)
\(558\) 0 0
\(559\) 47.4056 2.00504
\(560\) 0 0
\(561\) 5.69571 0.240473
\(562\) 0 0
\(563\) −0.509966 −0.0214925 −0.0107463 0.999942i \(-0.503421\pi\)
−0.0107463 + 0.999942i \(0.503421\pi\)
\(564\) 0 0
\(565\) −1.51217 −0.0636175
\(566\) 0 0
\(567\) −17.1359 −0.719641
\(568\) 0 0
\(569\) 38.1228 1.59819 0.799096 0.601204i \(-0.205312\pi\)
0.799096 + 0.601204i \(0.205312\pi\)
\(570\) 0 0
\(571\) 12.8778 0.538921 0.269460 0.963011i \(-0.413155\pi\)
0.269460 + 0.963011i \(0.413155\pi\)
\(572\) 0 0
\(573\) 10.7375 0.448567
\(574\) 0 0
\(575\) 0.221746 0.00924746
\(576\) 0 0
\(577\) 1.07288 0.0446646 0.0223323 0.999751i \(-0.492891\pi\)
0.0223323 + 0.999751i \(0.492891\pi\)
\(578\) 0 0
\(579\) −20.6848 −0.859632
\(580\) 0 0
\(581\) −65.6238 −2.72253
\(582\) 0 0
\(583\) 0.597747 0.0247561
\(584\) 0 0
\(585\) −2.89347 −0.119630
\(586\) 0 0
\(587\) −17.1600 −0.708270 −0.354135 0.935194i \(-0.615225\pi\)
−0.354135 + 0.935194i \(0.615225\pi\)
\(588\) 0 0
\(589\) −9.63624 −0.397054
\(590\) 0 0
\(591\) 21.7005 0.892640
\(592\) 0 0
\(593\) −13.0479 −0.535813 −0.267906 0.963445i \(-0.586332\pi\)
−0.267906 + 0.963445i \(0.586332\pi\)
\(594\) 0 0
\(595\) 9.07722 0.372130
\(596\) 0 0
\(597\) −2.85773 −0.116959
\(598\) 0 0
\(599\) 33.6344 1.37427 0.687133 0.726532i \(-0.258869\pi\)
0.687133 + 0.726532i \(0.258869\pi\)
\(600\) 0 0
\(601\) −36.7673 −1.49977 −0.749884 0.661569i \(-0.769891\pi\)
−0.749884 + 0.661569i \(0.769891\pi\)
\(602\) 0 0
\(603\) −13.8305 −0.563221
\(604\) 0 0
\(605\) 5.31779 0.216199
\(606\) 0 0
\(607\) −4.64360 −0.188478 −0.0942390 0.995550i \(-0.530042\pi\)
−0.0942390 + 0.995550i \(0.530042\pi\)
\(608\) 0 0
\(609\) 14.8956 0.603600
\(610\) 0 0
\(611\) 23.3862 0.946105
\(612\) 0 0
\(613\) 8.43345 0.340624 0.170312 0.985390i \(-0.445522\pi\)
0.170312 + 0.985390i \(0.445522\pi\)
\(614\) 0 0
\(615\) 7.77205 0.313399
\(616\) 0 0
\(617\) −48.5749 −1.95555 −0.977777 0.209650i \(-0.932768\pi\)
−0.977777 + 0.209650i \(0.932768\pi\)
\(618\) 0 0
\(619\) 43.6820 1.75573 0.877863 0.478912i \(-0.158969\pi\)
0.877863 + 0.478912i \(0.158969\pi\)
\(620\) 0 0
\(621\) 0.266059 0.0106766
\(622\) 0 0
\(623\) −33.7036 −1.35031
\(624\) 0 0
\(625\) 20.4022 0.816090
\(626\) 0 0
\(627\) 1.62001 0.0646970
\(628\) 0 0
\(629\) −1.63690 −0.0652675
\(630\) 0 0
\(631\) −6.36137 −0.253242 −0.126621 0.991951i \(-0.540413\pi\)
−0.126621 + 0.991951i \(0.540413\pi\)
\(632\) 0 0
\(633\) −4.17359 −0.165885
\(634\) 0 0
\(635\) −1.74729 −0.0693392
\(636\) 0 0
\(637\) −59.3566 −2.35179
\(638\) 0 0
\(639\) 17.7913 0.703814
\(640\) 0 0
\(641\) 13.3986 0.529211 0.264605 0.964357i \(-0.414758\pi\)
0.264605 + 0.964357i \(0.414758\pi\)
\(642\) 0 0
\(643\) 33.0080 1.30171 0.650854 0.759203i \(-0.274411\pi\)
0.650854 + 0.759203i \(0.274411\pi\)
\(644\) 0 0
\(645\) 8.46039 0.333127
\(646\) 0 0
\(647\) −45.2131 −1.77751 −0.888754 0.458384i \(-0.848429\pi\)
−0.888754 + 0.458384i \(0.848429\pi\)
\(648\) 0 0
\(649\) 2.23822 0.0878580
\(650\) 0 0
\(651\) 58.9020 2.30855
\(652\) 0 0
\(653\) −3.31559 −0.129749 −0.0648746 0.997893i \(-0.520665\pi\)
−0.0648746 + 0.997893i \(0.520665\pi\)
\(654\) 0 0
\(655\) −2.63774 −0.103065
\(656\) 0 0
\(657\) −6.23894 −0.243404
\(658\) 0 0
\(659\) 29.2922 1.14106 0.570531 0.821276i \(-0.306737\pi\)
0.570531 + 0.821276i \(0.306737\pi\)
\(660\) 0 0
\(661\) 21.1065 0.820948 0.410474 0.911872i \(-0.365363\pi\)
0.410474 + 0.911872i \(0.365363\pi\)
\(662\) 0 0
\(663\) 19.3415 0.751163
\(664\) 0 0
\(665\) 2.58180 0.100118
\(666\) 0 0
\(667\) −0.115293 −0.00446415
\(668\) 0 0
\(669\) 20.8864 0.807514
\(670\) 0 0
\(671\) 1.51635 0.0585380
\(672\) 0 0
\(673\) 36.9826 1.42557 0.712787 0.701381i \(-0.247433\pi\)
0.712787 + 0.701381i \(0.247433\pi\)
\(674\) 0 0
\(675\) 26.3573 1.01449
\(676\) 0 0
\(677\) 6.64796 0.255502 0.127751 0.991806i \(-0.459224\pi\)
0.127751 + 0.991806i \(0.459224\pi\)
\(678\) 0 0
\(679\) −52.1272 −2.00046
\(680\) 0 0
\(681\) −27.4856 −1.05325
\(682\) 0 0
\(683\) 30.0248 1.14887 0.574434 0.818551i \(-0.305222\pi\)
0.574434 + 0.818551i \(0.305222\pi\)
\(684\) 0 0
\(685\) 5.05971 0.193322
\(686\) 0 0
\(687\) 19.2673 0.735094
\(688\) 0 0
\(689\) 2.02983 0.0773305
\(690\) 0 0
\(691\) 19.5241 0.742734 0.371367 0.928486i \(-0.378889\pi\)
0.371367 + 0.928486i \(0.378889\pi\)
\(692\) 0 0
\(693\) 7.02724 0.266943
\(694\) 0 0
\(695\) 8.66485 0.328677
\(696\) 0 0
\(697\) 36.8681 1.39648
\(698\) 0 0
\(699\) 25.7940 0.975619
\(700\) 0 0
\(701\) 6.27584 0.237035 0.118518 0.992952i \(-0.462186\pi\)
0.118518 + 0.992952i \(0.462186\pi\)
\(702\) 0 0
\(703\) −0.465578 −0.0175596
\(704\) 0 0
\(705\) 4.17369 0.157190
\(706\) 0 0
\(707\) 67.7343 2.54741
\(708\) 0 0
\(709\) −28.4572 −1.06873 −0.534366 0.845253i \(-0.679450\pi\)
−0.534366 + 0.845253i \(0.679450\pi\)
\(710\) 0 0
\(711\) −1.24526 −0.0467007
\(712\) 0 0
\(713\) −0.455905 −0.0170738
\(714\) 0 0
\(715\) −2.84166 −0.106272
\(716\) 0 0
\(717\) 24.7527 0.924408
\(718\) 0 0
\(719\) −41.8197 −1.55961 −0.779806 0.626021i \(-0.784682\pi\)
−0.779806 + 0.626021i \(0.784682\pi\)
\(720\) 0 0
\(721\) 15.0433 0.560241
\(722\) 0 0
\(723\) −15.5808 −0.579458
\(724\) 0 0
\(725\) −11.4216 −0.424187
\(726\) 0 0
\(727\) 38.3026 1.42056 0.710282 0.703918i \(-0.248568\pi\)
0.710282 + 0.703918i \(0.248568\pi\)
\(728\) 0 0
\(729\) 26.9724 0.998976
\(730\) 0 0
\(731\) 40.1334 1.48439
\(732\) 0 0
\(733\) −20.6454 −0.762554 −0.381277 0.924461i \(-0.624516\pi\)
−0.381277 + 0.924461i \(0.624516\pi\)
\(734\) 0 0
\(735\) −10.5933 −0.390738
\(736\) 0 0
\(737\) −13.5828 −0.500330
\(738\) 0 0
\(739\) 19.2867 0.709473 0.354736 0.934966i \(-0.384571\pi\)
0.354736 + 0.934966i \(0.384571\pi\)
\(740\) 0 0
\(741\) 5.50124 0.202093
\(742\) 0 0
\(743\) 7.45431 0.273472 0.136736 0.990608i \(-0.456339\pi\)
0.136736 + 0.990608i \(0.456339\pi\)
\(744\) 0 0
\(745\) 2.75911 0.101086
\(746\) 0 0
\(747\) 17.7094 0.647953
\(748\) 0 0
\(749\) −42.5698 −1.55547
\(750\) 0 0
\(751\) 49.1641 1.79402 0.897012 0.442006i \(-0.145733\pi\)
0.897012 + 0.442006i \(0.145733\pi\)
\(752\) 0 0
\(753\) −1.10686 −0.0403361
\(754\) 0 0
\(755\) 7.14684 0.260100
\(756\) 0 0
\(757\) −51.8672 −1.88514 −0.942572 0.334003i \(-0.891601\pi\)
−0.942572 + 0.334003i \(0.891601\pi\)
\(758\) 0 0
\(759\) 0.0766450 0.00278204
\(760\) 0 0
\(761\) 30.0459 1.08916 0.544582 0.838708i \(-0.316688\pi\)
0.544582 + 0.838708i \(0.316688\pi\)
\(762\) 0 0
\(763\) 27.1787 0.983934
\(764\) 0 0
\(765\) −2.44960 −0.0885656
\(766\) 0 0
\(767\) 7.60058 0.274441
\(768\) 0 0
\(769\) 9.01146 0.324962 0.162481 0.986712i \(-0.448050\pi\)
0.162481 + 0.986712i \(0.448050\pi\)
\(770\) 0 0
\(771\) −11.7148 −0.421897
\(772\) 0 0
\(773\) 10.1037 0.363406 0.181703 0.983353i \(-0.441839\pi\)
0.181703 + 0.983353i \(0.441839\pi\)
\(774\) 0 0
\(775\) −45.1646 −1.62236
\(776\) 0 0
\(777\) 2.84587 0.102095
\(778\) 0 0
\(779\) 10.4863 0.375710
\(780\) 0 0
\(781\) 17.4727 0.625223
\(782\) 0 0
\(783\) −13.7040 −0.489740
\(784\) 0 0
\(785\) −7.91529 −0.282509
\(786\) 0 0
\(787\) 49.6757 1.77075 0.885373 0.464881i \(-0.153903\pi\)
0.885373 + 0.464881i \(0.153903\pi\)
\(788\) 0 0
\(789\) −14.9757 −0.533148
\(790\) 0 0
\(791\) 12.4712 0.443425
\(792\) 0 0
\(793\) 5.14923 0.182855
\(794\) 0 0
\(795\) 0.362261 0.0128481
\(796\) 0 0
\(797\) 34.2213 1.21218 0.606091 0.795395i \(-0.292737\pi\)
0.606091 + 0.795395i \(0.292737\pi\)
\(798\) 0 0
\(799\) 19.7987 0.700427
\(800\) 0 0
\(801\) 9.09534 0.321368
\(802\) 0 0
\(803\) −6.12721 −0.216225
\(804\) 0 0
\(805\) 0.122149 0.00430518
\(806\) 0 0
\(807\) 21.5409 0.758276
\(808\) 0 0
\(809\) −29.4763 −1.03633 −0.518166 0.855280i \(-0.673385\pi\)
−0.518166 + 0.855280i \(0.673385\pi\)
\(810\) 0 0
\(811\) −10.4739 −0.367789 −0.183895 0.982946i \(-0.558871\pi\)
−0.183895 + 0.982946i \(0.558871\pi\)
\(812\) 0 0
\(813\) 12.8253 0.449804
\(814\) 0 0
\(815\) −5.29566 −0.185499
\(816\) 0 0
\(817\) 11.4150 0.399360
\(818\) 0 0
\(819\) 23.8631 0.833846
\(820\) 0 0
\(821\) −41.9290 −1.46333 −0.731666 0.681663i \(-0.761257\pi\)
−0.731666 + 0.681663i \(0.761257\pi\)
\(822\) 0 0
\(823\) −37.5051 −1.30735 −0.653674 0.756777i \(-0.726773\pi\)
−0.653674 + 0.756777i \(0.726773\pi\)
\(824\) 0 0
\(825\) 7.59290 0.264351
\(826\) 0 0
\(827\) 14.2808 0.496592 0.248296 0.968684i \(-0.420129\pi\)
0.248296 + 0.968684i \(0.420129\pi\)
\(828\) 0 0
\(829\) 14.9066 0.517726 0.258863 0.965914i \(-0.416652\pi\)
0.258863 + 0.965914i \(0.416652\pi\)
\(830\) 0 0
\(831\) 28.9499 1.00426
\(832\) 0 0
\(833\) −50.2510 −1.74110
\(834\) 0 0
\(835\) 5.40539 0.187061
\(836\) 0 0
\(837\) −54.1899 −1.87308
\(838\) 0 0
\(839\) 48.1732 1.66312 0.831562 0.555432i \(-0.187447\pi\)
0.831562 + 0.555432i \(0.187447\pi\)
\(840\) 0 0
\(841\) −23.0616 −0.795227
\(842\) 0 0
\(843\) −10.2172 −0.351899
\(844\) 0 0
\(845\) −2.37610 −0.0817404
\(846\) 0 0
\(847\) −43.8570 −1.50695
\(848\) 0 0
\(849\) −15.2867 −0.524640
\(850\) 0 0
\(851\) −0.0220272 −0.000755081 0
\(852\) 0 0
\(853\) 15.1947 0.520258 0.260129 0.965574i \(-0.416235\pi\)
0.260129 + 0.965574i \(0.416235\pi\)
\(854\) 0 0
\(855\) −0.696732 −0.0238277
\(856\) 0 0
\(857\) 46.1113 1.57513 0.787567 0.616229i \(-0.211341\pi\)
0.787567 + 0.616229i \(0.211341\pi\)
\(858\) 0 0
\(859\) 13.3282 0.454753 0.227377 0.973807i \(-0.426985\pi\)
0.227377 + 0.973807i \(0.426985\pi\)
\(860\) 0 0
\(861\) −64.0979 −2.18445
\(862\) 0 0
\(863\) −34.6256 −1.17867 −0.589335 0.807889i \(-0.700610\pi\)
−0.589335 + 0.807889i \(0.700610\pi\)
\(864\) 0 0
\(865\) −11.6710 −0.396827
\(866\) 0 0
\(867\) −6.14489 −0.208691
\(868\) 0 0
\(869\) −1.22296 −0.0414859
\(870\) 0 0
\(871\) −46.1247 −1.56287
\(872\) 0 0
\(873\) 14.0672 0.476102
\(874\) 0 0
\(875\) 25.0098 0.845485
\(876\) 0 0
\(877\) 3.00362 0.101425 0.0507126 0.998713i \(-0.483851\pi\)
0.0507126 + 0.998713i \(0.483851\pi\)
\(878\) 0 0
\(879\) −42.9296 −1.44798
\(880\) 0 0
\(881\) 8.88260 0.299262 0.149631 0.988742i \(-0.452191\pi\)
0.149631 + 0.988742i \(0.452191\pi\)
\(882\) 0 0
\(883\) −2.55538 −0.0859954 −0.0429977 0.999075i \(-0.513691\pi\)
−0.0429977 + 0.999075i \(0.513691\pi\)
\(884\) 0 0
\(885\) 1.35646 0.0455970
\(886\) 0 0
\(887\) −36.1162 −1.21266 −0.606332 0.795211i \(-0.707360\pi\)
−0.606332 + 0.795211i \(0.707360\pi\)
\(888\) 0 0
\(889\) 14.4103 0.483307
\(890\) 0 0
\(891\) 4.54153 0.152147
\(892\) 0 0
\(893\) 5.63127 0.188443
\(894\) 0 0
\(895\) −4.72086 −0.157801
\(896\) 0 0
\(897\) 0.260272 0.00869022
\(898\) 0 0
\(899\) 23.4824 0.783183
\(900\) 0 0
\(901\) 1.71845 0.0572498
\(902\) 0 0
\(903\) −69.7748 −2.32196
\(904\) 0 0
\(905\) 8.29612 0.275772
\(906\) 0 0
\(907\) −21.0396 −0.698609 −0.349305 0.937009i \(-0.613582\pi\)
−0.349305 + 0.937009i \(0.613582\pi\)
\(908\) 0 0
\(909\) −18.2790 −0.606275
\(910\) 0 0
\(911\) −8.98196 −0.297585 −0.148793 0.988868i \(-0.547539\pi\)
−0.148793 + 0.988868i \(0.547539\pi\)
\(912\) 0 0
\(913\) 17.3923 0.575600
\(914\) 0 0
\(915\) 0.918974 0.0303803
\(916\) 0 0
\(917\) 21.7541 0.718383
\(918\) 0 0
\(919\) 32.1506 1.06055 0.530275 0.847825i \(-0.322089\pi\)
0.530275 + 0.847825i \(0.322089\pi\)
\(920\) 0 0
\(921\) 21.3090 0.702156
\(922\) 0 0
\(923\) 59.3340 1.95300
\(924\) 0 0
\(925\) −2.18214 −0.0717483
\(926\) 0 0
\(927\) −4.05962 −0.133335
\(928\) 0 0
\(929\) −18.5445 −0.608426 −0.304213 0.952604i \(-0.598393\pi\)
−0.304213 + 0.952604i \(0.598393\pi\)
\(930\) 0 0
\(931\) −14.2927 −0.468425
\(932\) 0 0
\(933\) −2.47662 −0.0810808
\(934\) 0 0
\(935\) −2.40574 −0.0786760
\(936\) 0 0
\(937\) −50.9837 −1.66556 −0.832782 0.553601i \(-0.813253\pi\)
−0.832782 + 0.553601i \(0.813253\pi\)
\(938\) 0 0
\(939\) 29.1085 0.949919
\(940\) 0 0
\(941\) 14.3782 0.468715 0.234358 0.972150i \(-0.424701\pi\)
0.234358 + 0.972150i \(0.424701\pi\)
\(942\) 0 0
\(943\) 0.496121 0.0161559
\(944\) 0 0
\(945\) 14.5189 0.472300
\(946\) 0 0
\(947\) −35.8756 −1.16580 −0.582901 0.812543i \(-0.698082\pi\)
−0.582901 + 0.812543i \(0.698082\pi\)
\(948\) 0 0
\(949\) −20.8068 −0.675419
\(950\) 0 0
\(951\) 36.1692 1.17287
\(952\) 0 0
\(953\) −26.1923 −0.848450 −0.424225 0.905557i \(-0.639454\pi\)
−0.424225 + 0.905557i \(0.639454\pi\)
\(954\) 0 0
\(955\) −4.53529 −0.146758
\(956\) 0 0
\(957\) −3.94778 −0.127614
\(958\) 0 0
\(959\) −41.7286 −1.34749
\(960\) 0 0
\(961\) 61.8571 1.99539
\(962\) 0 0
\(963\) 11.4880 0.370196
\(964\) 0 0
\(965\) 8.73680 0.281248
\(966\) 0 0
\(967\) 40.3371 1.29715 0.648577 0.761149i \(-0.275365\pi\)
0.648577 + 0.761149i \(0.275365\pi\)
\(968\) 0 0
\(969\) 4.65733 0.149615
\(970\) 0 0
\(971\) −56.7807 −1.82218 −0.911089 0.412209i \(-0.864757\pi\)
−0.911089 + 0.412209i \(0.864757\pi\)
\(972\) 0 0
\(973\) −71.4610 −2.29094
\(974\) 0 0
\(975\) 25.7840 0.825750
\(976\) 0 0
\(977\) −1.41925 −0.0454057 −0.0227029 0.999742i \(-0.507227\pi\)
−0.0227029 + 0.999742i \(0.507227\pi\)
\(978\) 0 0
\(979\) 8.93246 0.285483
\(980\) 0 0
\(981\) −7.33451 −0.234173
\(982\) 0 0
\(983\) −29.7135 −0.947713 −0.473856 0.880602i \(-0.657138\pi\)
−0.473856 + 0.880602i \(0.657138\pi\)
\(984\) 0 0
\(985\) −9.16580 −0.292047
\(986\) 0 0
\(987\) −34.4214 −1.09565
\(988\) 0 0
\(989\) 0.540060 0.0171729
\(990\) 0 0
\(991\) 32.4649 1.03128 0.515641 0.856805i \(-0.327554\pi\)
0.515641 + 0.856805i \(0.327554\pi\)
\(992\) 0 0
\(993\) −27.3702 −0.868567
\(994\) 0 0
\(995\) 1.20704 0.0382657
\(996\) 0 0
\(997\) 7.18573 0.227574 0.113787 0.993505i \(-0.463702\pi\)
0.113787 + 0.993505i \(0.463702\pi\)
\(998\) 0 0
\(999\) −2.61820 −0.0828362
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6004.2.a.f.1.17 25
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6004.2.a.f.1.17 25 1.1 even 1 trivial